[MUSIC].
Welcome back to Calculus One, and Welcome to Week 13 of our time together.
Well, we're almost to the end of the course.
Soon we will be posting the final exam, which will give you a few weeks to work
on that. But I wanted to say a little bit about
where we are. Last week we looked at how
anti-differentiation gives us the fundamental theorem of calculus, which
lets us evaluate integrals way more easily than by going back to the
definition in terms of Riemann sums. This week we continue on that quest,
we're going to look at u-substitution which is one really great technique for
actually finding some complicated antiderivatives.
Now, in light of this, I also want to say a little about how the course as a whole
has been organized. Remember way back at the beginning of the
course, we gave the definition of limit in terms of epsilons and deltas, but very
quickly, we abandoned those epsilons and deltas, and we gave a bunch of techniques
for evaluating limits. Things like the the limit of a sum is the
sum of the limits that the limits exist. And then, once we had easier techniques
for playing around with limits, we get some applications of limits.
A big one, being the limit of a difference quotient, which was the
definition of the derivative. But we didn't usually calculate the
derivative by going back to that definition.
Usually we used our techniques of differentiation like, the derivative is
of a sum is the sum of derivatives to make calculating derivatives way easier.
And then we did some applications of differentiations, right, things like
optimization problems, related rates problems, and we then did definition of
integration, right, in terms of Riemann sums.
But those Riemann sums were way to complicated, so we very quickly abandoned
Riemann sums for a, a, better technique Involving the fundamental theorem of
calculus. In this week, with use substitution, we
learned another technique for finding anti-derivatives, that is another
technique for evaluating definite integrals.
And in the last couple weeks, we're going to look at some more techniques and
some applications of integration. There's not much time left, hang in
there. We're all going to make it to the end
together.